Solving a parabolic equation

Let us set some global options for all code chunks in this document.

# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = TRUE,    
  # Disable warnings printed by R code chunks
  warning = TRUE,    
  # Show R code within code chunks in output
  echo = TRUE,        
  # Include both R code and its results in output
  include = TRUE,     
  # Evaluate R code chunks
  eval = TRUE,       
  # Enable caching of R code chunks for faster rendering
  cache = FALSE,      
  # Align figures in the center of the output
  fig.align = "center",
  # Enable retina display for high-resolution figures
  retina = 2,
  # Show errors in the output instead of stopping rendering
  error = TRUE,
  # Do not collapse code and output into a single block
  collapse = FALSE
)
# Start the figure counter
fig_count <- 0
# Define the captioner function
captioner <- function(caption) {
  fig_count <<- fig_count + 1
  paste0("Figure ", fig_count, ": ", caption)
}
# Define the function to truncate a number to two decimal places
truncate_to_two <- function(x) {
  floor(x * 100) / 100
}

m1table <- rSPDE:::m1table
m2table <- rSPDE:::m2table
m3table <- rSPDE:::m3table
m4table <- rSPDE:::m4table

# install.packages("INLA",repos=c(getOption("repos"),INLA="https://inla.r-inla-download.org/R/testing"), dep=TRUE)
# inla.upgrade(testing = TRUE)
# remotes::install_github("inlabru-org/inlabru", ref = "devel")
# remotes::install_github("davidbolin/rspde", ref = "devel")
# remotes::install_github("davidbolin/metricgraph", ref = "devel")
library(INLA)

## Loading required package: Matrix

## This is INLA_25.05.01 built 2025-05-01 18:43:33 UTC.
##  - See www.r-inla.org/contact-us for how to get help.
##  - List available models/likelihoods/etc with inla.list.models()
##  - Use inla.doc(<NAME>) to access documentation
##  - Consider upgrading R-INLA to testing[25.05.07] or stable[24.12.11].

library(inlabru)

## Loading required package: fmesher

library(rSPDE)

## This is rSPDE 2.5.1
## - See https://davidbolin.github.io/rSPDE for vignettes and manuals.

library(MetricGraph)

## This is MetricGraph 1.4.1
## - See https://davidbolin.github.io/MetricGraph for vignettes and manuals.

## 
## Attaching package: 'MetricGraph'

## The following object is masked from 'package:stats':
## 
##     filter

library(grateful)

library(plotly)

## Loading required package: ggplot2

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

Utilitary functions

# For each m and beta, this function returns c_m/b_{m+1} and the roots of rb and rc
my.get.roots <- function(order, beta) {
  mt <- get(paste0("m", order, "table"))
  rb <- rep(0, order + 1)
  rc <- rep(0, order)
  if(order == 1) {
      rc = approx(mt$beta, mt[[paste0("rc")]], beta)$y
    } else {
      rc = sapply(1:order, function(i) {
        approx(mt$beta, mt[[paste0("rc.", i)]], beta)$y
      })
    }
    rb = sapply(1:(order+1), function(i) {
      approx(mt$beta, mt[[paste0("rb.", i)]], xout = beta)$y
    })
    factor = approx(mt$beta, mt$factor, xout = beta)$y
  return(list(rb = rb, rc = rc, factor = factor))
}

# Function the polynomial coefficients in increasing order like a+bx+cx^2+...
poly.from.roots <- function(roots) {
  
  coef <- 1
  for (r in roots) {coef <- convolve(coef, c(1, -r), type = "open")}
  return(coef)
}
# Function to compute the partial fraction parameters
compute.partial.fraction.param <- function(factor, pr_roots, pl_roots, time_step, scaling) {
  
  pr_coef <- c(0, poly.from.roots(pr_roots)) 
  pl_coef <- poly.from.roots(pl_roots) 
  factor_pr_coef <- pr_coef
  pr_plus_pl_coef <- factor_pr_coef + ((scaling * time_step)/factor) * pl_coef
  res <- gsignal::residue(factor_pr_coef, pr_plus_pl_coef)
  return(list(r = res$r, p = res$p, k = res$k)) 
}
# Function to compute the fractional operator
my.fractional.operators.frac <- function(L, beta, C, scale.factor, m = 1, time_step) {
  
  C <- Matrix::Diagonal(dim(C)[1], rowSums(C)) 
  Ci <- Matrix::Diagonal(dim(C)[1], 1 / rowSums(C)) 
  I <- Matrix::Diagonal(dim(C)[1])
  L <- L / scale.factor 
  LCi <- L %*% Ci
  if(beta == 1){
    L <- L * scale.factor^beta
    return(list(Ci = Ci, C = C, LCi = LCi, L = L, m = m, beta = beta, LHS = C + time_step * L))
  } else {
    scaling <- scale.factor^beta
    roots <- my.get.roots(m, beta)
    poles_rs_k <- compute.partial.fraction.param(roots$factor, roots$rc, roots$rb, time_step, scaling)

    partial_fraction_terms <- list()
    for (i in 1:(m+1)) {partial_fraction_terms[[i]] <- (LCi - poles_rs_k$p[i] * I)/poles_rs_k$r[i]}
    partial_fraction_terms[[m+2]] <- ifelse(is.null(poles_rs_k$k), 0, poles_rs_k$k) * I
    return(list(Ci = Ci, C = C, LCi = LCi, L = L, m = m, beta = beta, partial_fraction_terms = partial_fraction_terms))
  }
}
# Function to solve the iteration
my.solver.frac <- function(obj, v){
  beta <- obj$beta
  m <- obj$m
  C <- obj$C
  Ci <- obj$Ci
  if (beta == 1){
    return(solve(obj$LHS, v))
  } else {
    partial_fraction_terms <- obj$partial_fraction_terms
    output <- partial_fraction_terms[[m+2]] %*% v
    for (i in 1:(m+1)) {output <- output + solve(partial_fraction_terms[[i]], v)}
    return(Ci %*% output)
  }
}
# Function to build a tadpole graph and create a mesh
gets.graph.tadpole <- function(h){
  edge1 <- rbind(c(0,0),c(1,0))
  theta <- seq(from=-pi,to=pi,length.out = 10000)
  edge2 <- cbind(1+1/pi+cos(theta)/pi,sin(theta)/pi)
  edges <- list(edge1, edge2)
  graph <- metric_graph$new(edges = edges)
  graph$set_manual_edge_lengths(edge_lengths = c(1,2))
  graph$build_mesh(h = h)
  return(graph)
}
# Function to compute the eigenfunctions 
tadpole.eig <- function(k,graph){
x1 <- c(0,graph$get_edge_lengths()[1]*graph$mesh$PtE[graph$mesh$PtE[,1]==1,2]) 
x2 <- c(0,graph$get_edge_lengths()[2]*graph$mesh$PtE[graph$mesh$PtE[,1]==2,2]) 

if(k==0){ 
  f.e1 <- rep(1,length(x1)) 
  f.e2 <- rep(1,length(x2)) 
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  f = list(phi=f1/sqrt(3)) 
  
} else {
  f.e1 <- -2*sin(pi*k*1/2)*cos(pi*k*x1/2) 
  f.e2 <- sin(pi*k*x2/2)                  
  
  f1 = c(f.e1[1],f.e2[1],f.e1[-1], f.e2[-1]) 
  
  if((k %% 2)==1){ 
    f = list(phi=f1/sqrt(3)) 
  } else { 
    f.e1 <- (-1)^{k/2}*cos(pi*k*x1/2)
    f.e2 <- cos(pi*k*x2/2)
    f2 = c(f.e1[1],f.e2[1],f.e1[-1],f.e2[-1]) 
    f <- list(phi=f1,psi=f2/sqrt(3/2))
  }
}
return(f)
}
# Function to order the vertices for plotting
plotting.order <- function(v, graph){
  edge_number <- graph$mesh$VtE[, 1]
  pos <- sum(edge_number == 1)+1
  return(c(v[1], v[3:pos], v[2], v[(pos+1):length(v)], v[2]))
}
# Function to plot in 3D
graph.plotter.3d <- function(graph, U_true, U_approx1, U_approx2, time_seq, time_step){
  x <- graph$mesh$V[, 1]
  y <- graph$mesh$V[, 2]
  x <- plotting.order(x, graph)
  y <- plotting.order(y, graph)
  weights <- graph$mesh$weights
  
  cumsum1 <- sqrt(time_step * cumsum(t(weights) %*% (U_true - U_approx1)^2))
  cumsum2 <- sqrt(time_step * cumsum(t(weights) %*%(U_true - U_approx2)^2))
  cumsum3 <- sqrt(time_step * cumsum(t(weights) %*% (U_approx1 - U_approx2)^2))
  
  U_true <- apply(U_true, 2, plotting.order, graph = graph)
  U_approx1 <- apply(U_approx1, 2, plotting.order, graph = graph)
  U_approx2 <- apply(U_approx2, 2, plotting.order, graph = graph)
  
  plot_data <- data.frame(
  x = rep(x, times = ncol(U_true)),
  y = rep(y, times = ncol(U_true)),
  z_true = as.vector(U_true),
  z_approx1 = as.vector(U_approx1),
  z_approx2 = as.vector(U_approx2),
  frame = rep(time_seq, each = length(x)))
  
  x_range <- range(x)
  y_range <- range(y)
  z_range <- range(c(U_true, U_approx1, U_approx2))
  
  p <- plot_ly(plot_data, frame = ~frame) %>%
  add_trace(x = ~x, y = ~y, z = ~z_true, type = "scatter3d", mode = "lines", name = "T",
            line = list(color = "green", width = 2)) %>%
  add_trace(x = ~x, y = ~y, z = ~z_approx1, type = "scatter3d", mode = "lines", name = "A1",
            line = list(color = "red", width = 2)) %>%
  add_trace(x = ~x, y = ~y, z = ~z_approx2, type = "scatter3d", mode = "lines", name = "A2", 
            line = list(color = "blue", width = 2)) %>%
  layout(
    scene = list(
      xaxis = list(title = "x", range = x_range),
      yaxis = list(title = "y", range = y_range),
      zaxis = list(title = "z", range = z_range),
      aspectratio = list(x = 2.4, y = 1.2, z = 1.2),
      camera = list(eye = list(x = 1.5, y = 1.5, z = 1), center = list(x = 0, y = 0, z = 0))),
    updatemenus = list(list(type = "buttons", showactive = FALSE,
                            buttons = list(
          list(label = "Play", method = "animate",
               args = list(NULL, list(frame = list(duration = 100, redraw = TRUE), fromcurrent = TRUE))),
          list(label = "Pause", method = "animate",
               args = list(NULL, list(mode = "immediate", frame = list(duration = 0), redraw = FALSE)))
        )
      )
    ),
    title = "Time: 0"
  ) %>% plotly_build()
  for (i in seq_along(p$x$frames)) {
    t <- time_seq[i]
    err1 <- signif(cumsum1[i], 4)
    err2 <- signif(cumsum2[i], 4)
    err3 <- signif(cumsum3[i], 4)
    p$x$frames[[i]]$layout <- list(title = paste0("Time: ", t, " | cum E=T-A1: ", err1, " | cum E=T-A2: ", err2, " | cum E=A1-A2: ", err3))
  }
  return(p)
}
# Function to plot the error at each time step
error.at.each.time.plotter <- function(graph, U_true, U_approx1, U_approx2, time_seq, time_step) {
  weights <- graph$mesh$weights
  error_at_each_time1 <- t(weights) %*% (U_true - U_approx1)^2
  error_at_each_time2 <- t(weights) %*% (U_true - U_approx2)^2
  error_between_both_approx <- t(weights) %*% (U_approx1 - U_approx2)^2
  
  error1 <- sqrt(as.double(t(weights) %*% (U_true - U_approx1)^2 %*% rep(time_step, ncol(U_true))))
  error2 <- sqrt(as.double(t(weights) %*% (U_true - U_approx2)^2 %*% rep(time_step, ncol(U_true))))
  errorb <- sqrt(as.double(t(weights) %*% (U_approx1 - U_approx2)^2 %*% rep(time_step, ncol(U_true))))
  
  fig <- plot_ly() %>% 
  add_trace(
  x = ~time_seq, y = ~error_at_each_time1, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'red', width = 2),
  marker = list(color = 'red', size = 4),
  name = paste0("E=T-A1: ", sprintf("%.3e", error1))
) %>% 
  add_trace(
  x = ~time_seq, y = ~error_at_each_time2, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'blue', width = 2, dash = "dot"),
  marker = list(color = 'blue', size = 4),
  name = paste0("E=T-A2: ", sprintf("%.3e", error2))
) %>% 
  add_trace(
  x = ~time_seq, y = ~error_between_both_approx, type = 'scatter', mode = 'lines+markers',
  line = list(color = 'orange', width = 2),
  marker = list(color = 'orange', size = 4),
  name = paste0("E=A1-A2: ", sprintf("%.3e", errorb))
) %>% 
  layout(
  title = "Error at Each Time Step",
  xaxis = list(title = "Error at Each Time Step"),
  yaxis = list(title = "Error"),
  legend = list(x = 0.1, y = 0.9)
)
  return(fig)
}
# Function to compute the eigenvalues and eigenfunctions
gets.eigen.params <- function(N_finite = 4, kappa = 1, alpha = 0.5, graph){
  EIGENVAL_ALPHA <- NULL
  EIGENFUN <- NULL
  for (j in 0:N_finite) {
      lambda_j_alpha <- (kappa^2 + (j*pi/2)^2)^(alpha/2)
      e_j <- tadpole.eig(j,graph)$phi
      EIGENVAL_ALPHA <- c(EIGENVAL_ALPHA, lambda_j_alpha)         
      EIGENFUN <- cbind(EIGENFUN, e_j)
      if (j>0 && (j %% 2 == 0)) {
        lambda_j_alpha <- (kappa^2 + (j*pi/2)^2)^(alpha/2)
        e_j <- tadpole.eig(j,graph)$psi
        EIGENVAL_ALPHA <- c(EIGENVAL_ALPHA, lambda_j_alpha)         
        EIGENFUN <- cbind(EIGENFUN, e_j)
      }
  }
  return(list(EIGENVAL_ALPHA = EIGENVAL_ALPHA,
              EIGENFUN = EIGENFUN))
}

We want to solve the fractional diffusion equation \[\begin{equation} \label{eq:maineq} \partial_t u+(\kappa^2-\Delta_\Gamma)^{\alpha/2} u=f \text { on } \Gamma \times(0, T), \quad u(0)=u_0 \text { on } \Gamma, \end{equation}\] where \(u\) satisfies the Kirchhoff vertex conditions \[\begin{equation} \label{eq:Kcond} \left\{\phi\in C(\Gamma)\;\Big|\; \forall v\in V: \sum_{e\in\mathcal{E}_v}\partial_e \phi(v)=0 \right\} \end{equation}\] The solution is given by \[\begin{equation} \label{eq:sol_reprentation} u(s,t) = \displaystyle\sum_{j\in\mathbb{N}}e^{-\lambda^{\alpha/2}_jt}\left(u_0, e_j\right)_{L_2(\Gamma)}e_j(s) + \int_0^t \displaystyle\sum_{j\in\mathbb{N}}e^{-\lambda^{\alpha/2}_j(t-r)}\left(f(\cdot, r), e_j\right)_{L_2(\Gamma)}e_j(s)dr. \end{equation}\]

If we choose \(w_j\) and \(v_j\) and take the initial condition and the right hand side funciton as

\[\begin{equation} u_0(s) = \sum_{j=0}^{N} w_j e_j(s) \text{ and so } \left(u_0, e_j\right)_{L_2(\Gamma)} = w_j, \end{equation}\] \[\begin{equation} \text{In matrix notation: } \quad\boldsymbol{U}_0 = \boldsymbol{E}_h\boldsymbol{c}, \quad \boldsymbol{E}_h = \left[e_0, e_1, \ldots, e_{N}\right], \quad \boldsymbol{w} = \left[w_0, w_1, \ldots, w_{N}\right]^\top, \end{equation}\] \[\begin{equation} \text{In } \texttt{R}: \texttt{U_0 <- EIGENFUN %*% coeff_U_0} \end{equation}\] and \[\begin{equation} f(s,t) = \sum_{j=0}^{N} v_j e^{-\lambda^{\alpha/2}_jt} e_j(s) \text{ and so } \left(f(\cdot,r), e_j\right)_{L_2(\Gamma)} = v_j e^{-\lambda^{\alpha/2}_jr}, \end{equation}\] \[\begin{equation} \text{In matrix notation: } \quad\boldsymbol{f} = \boldsymbol{E}_h \boldsymbol{V}, \quad \boldsymbol{V}_{ji} = v_je^{-\lambda^{\alpha/2}_jt_i} \end{equation}\] \[\begin{equation} \text{In } \texttt{R}: \texttt{FF <- EIGENFUN %*% (coeff_FF * exp(-outer(EIGENVAL_ALPHA, time_seq)))} \end{equation}\] then the solution is given by \[\begin{align} u(s,t) &= \displaystyle\sum_{j=0}^{N}w_je^{-\lambda^{\alpha/2}_jt}e_j(s) + \int_0^t \displaystyle\sum_{j=0}^Ne^{-\lambda^{\alpha/2}_j(t-r)}v_j e^{-\lambda^{\alpha/2}_jr}e_j(s)dr\\ &= \displaystyle\sum_{j=0}^{N}w_je^{-\lambda^{\alpha/2}_jt}e_j(s) + t \displaystyle\sum_{j=0}^Nv_j e^{-\lambda^{\alpha/2}_jt}e_j(s)\\ &= \displaystyle\sum_{j=0}^{N}w_je^{-\lambda^{\alpha/2}_jt}e_j(s) + tf(s,t)\\ &= \displaystyle\sum_{j=0}^{N}(w_j+tv_j)e^{-\lambda^{\alpha/2}_jt}e_j(s) \end{align}\]

\[\begin{equation} \text{In matrix notation: } \quad\boldsymbol{U} =\boldsymbol{E}_h \boldsymbol{W} + (\boldsymbol{t}\boldsymbol{f}^\top)^\top, \quad \boldsymbol{W}_{ji} = w_je^{-\lambda^{\alpha/2}_jt_i},\quad \boldsymbol{t} = \left[t_0, t_1, \ldots, t_{K}\right]^\top \end{equation}\] \[\begin{equation} \text{In } \texttt{R}: \texttt{U_true <- EIGENFUN %*% (coeff_U_0 * exp(-outer(EIGENVAL_ALPHA, time_seq)))+ t(time_seq * t(FF))} \end{equation}\]

Numerical solution

From this, we arrive at the scheme

\[\begin{equation} (P_r^TC+\tau P_\ell^T)U^{k+1} = P_r^T (CU^k+\tau F^{k+1}) \label{eq:scheme} \end{equation}\] where (here we are using \(P_r\) and \(P_l\) the way they were constructed) \[\begin{equation} P_r = \prod_{i=1}^m \left(I-r_{1i}\dfrac{C^{-1}L}{\kappa^2}\right)\quad\text{and}\quad P_\ell = \dfrac{\kappa^{2\beta}}{\texttt{factor}}C\prod_{j=1}^{m+1} \left(I-r_{2j}\dfrac{C^{-1}L}{\kappa^2}\right) \end{equation}\] where \(\texttt{factor} = \dfrac{c_m}{b_{m+1}}\). Observe that \[\begin{equation} P_r^T = \prod_{i=1}^m \left(I-r_{1i}\dfrac{LC^{-1}}{\kappa^2}\right)\quad\text{and}\quad P_\ell^T = \dfrac{\kappa^{2\beta}}{\texttt{factor}}\prod_{j=1}^{m+1} \left(I-r_{2j}\dfrac{LC^{-1}}{\kappa^2}\right)\cdot C \end{equation}\] since \(C\), \(L\) and \(C^{-1}\) are symmetric and the factors in the product commute. Replacing these two in our scheme we get \[\begin{equation} \left(\prod_{i=1}^m \left(I-r_{1i}\dfrac{LC^{-1}}{\kappa^2}\right)+\dfrac{\tau \kappa^{2\beta}}{\texttt{factor}}\prod_{j=1}^{m+1} \left(I-r_{2j}\dfrac{LC^{-1}}{\kappa^2}\right)\right)CU^{k+1} = \prod_{i=1}^m \left(I-r_{1i}\dfrac{LC^{-1}}{\kappa^2}\right)\cdot (CU^k+\tau F^{k+1}) \end{equation}\] That is, \[\begin{equation} U^{k+1} = C^{-1}\left(\prod_{i=1}^m \left(I-r_{1i}\dfrac{LC^{-1}}{\kappa^2}\right)+\dfrac{\tau \kappa^{2\beta}}{\texttt{factor}}\prod_{j=1}^{m+1} \left(I-r_{2j}\dfrac{LC^{-1}}{\kappa^2}\right)\right)^{-1} \prod_{i=1}^m \left(I-r_{1i}\dfrac{LC^{-1}}{\kappa^2}\right)\cdot (CU^k+\tau F^{k+1}) \end{equation}\] We consider a partial fraction decomposition

\[\begin{equation} \dfrac{\prod_{i=1}^m (1-r_{1i}x)}{\prod_{i=1}^m (1-r_{1i}x)+\dfrac{\tau \kappa^{2\beta}}{\texttt{factor}} \prod_{j=1}^{m+1} (1-r_{2j}x)} \end{equation}\]

That is

\[\begin{equation} \sum_{k=1}^{m+1} \dfrac{a_k}{x-p_k} + r = \sum_{k=1}^{m+1} a_k(x-p_k)^{-1} + r \end{equation}\]

Therefore

\[\begin{equation} U^{k+1} = C^{-1}\left(\sum_{k=1}^{m+1} a_k\left( \dfrac{LC^{-1}}{\kappa^2}-p_kI\right)^{-1} + rI\right) (CU^k+\tau F^{k+1}) \end{equation}\]

h <- 0.01
T_final <- 2
time_step <- 0.01
kappa <- 0.01
alpha <- 2 # from 0.5 to 2
m = 1
beta <- alpha/2
N_finite = 4 # choose even
adjusted_N_finite <- N_finite + N_finite/2 + 1
# Coefficients for u_0 and f
coeff_U_0 <- 50*(1:adjusted_N_finite)^-1
coeff_U_0[-5] <- 0
coeff_FF <- rep(0, adjusted_N_finite)
coeff_FF[7] <- 10


time_seq <- seq(0, T_final, by = time_step)
graph <- gets.graph.tadpole(h = h)

## Starting graph creation...

## LongLat is set to FALSE

## Creating edges...

## Setting edge weights...

## Computing bounding box...

## Setting up edges

## Merging close vertices

## Total construction time: 0.28 secs

## Creating and updating vertices...

## Storing the initial graph...

## Computing the relative positions of the edges...

graph$compute_fem()
G <- graph$mesh$G
C <- graph$mesh$C
L <- kappa^2*C + G
I <- Matrix::Diagonal(nrow(C))

eigen_params <- gets.eigen.params(N_finite = N_finite, kappa = kappa, alpha = alpha, graph = graph)
EIGENVAL_ALPHA <- eigen_params$EIGENVAL_ALPHA
EIGENFUN <- eigen_params$EIGENFUN


U_0 <- EIGENFUN %*% coeff_U_0

U_true <- EIGENFUN %*% 
  outer(1:length(coeff_U_0), 
        1:length(time_seq), 
        function(i, j) (coeff_U_0[i] + coeff_FF[i]*time_seq[j]) * exp(-EIGENVAL_ALPHA[i] * time_seq[j]))

overkill_graph <- gets.graph.tadpole(h = 0.001)

## Starting graph creation...

## LongLat is set to FALSE

## Creating edges...

## Setting edge weights...

## Computing bounding box...

## Setting up edges

## Merging close vertices

## Total construction time: 0.21 secs

## Creating and updating vertices...

## Storing the initial graph...

## Computing the relative positions of the edges...

overkill_graph$compute_fem()
overkill_EIGENFUN <- gets.eigen.params(N_finite = N_finite, kappa = kappa, alpha = alpha, graph = overkill_graph)$EIGENFUN


INT_BASIS_EIGEN <- t(overkill_EIGENFUN) %*% 
  overkill_graph$mesh$C %*% 
  graph$fem_basis(overkill_graph$get_mesh_locations())


FF_approx <- t(INT_BASIS_EIGEN) %*% 
  outer(1:length(coeff_FF), 
        1:length(time_seq), 
        function(i, j) coeff_FF[i] * exp(-EIGENVAL_ALPHA[i] * time_seq[j]))

Solving it

op <- fractional.operators(L, beta, C, scale.factor = kappa^2, m = m)
Pl <- op$Pl
Pr <- op$Pr
Ci <- op$Ci
C <- op$C

Pr.apply.mult <- function(v){return(Pr.mult(op, v))}
Pl.apply.solve <- function(v){return(Pl.solve(op, v))}
PliC <- apply(C, 2, Pl.apply.solve) # Pl^-1C
# Precompute the LHS1 matrix
aux <- apply(PliC, 2, Pr.apply.mult) #PrPl^-1C
LHS <- aux + time_step * Matrix::Diagonal(nrow(C)) 

# Initialize U matrix to store solution at each time step
U_approx1 <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_approx1[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  # Compute the right-hand side for the second equation
  RHS <- aux %*% U_approx1[, k] + time_step * Pr.mult(op, Pl.solve(op, FF_approx[, k + 1]))
  U_approx1[, k + 1] <- as.matrix(solve(LHS, RHS))
}

{r}
op <- fractional.operators(L, beta, C, scale.factor = kappa^2, m = m)
Pl <- op$Pl
Pr <- op$Pr
Ci <- op$Ci
C <- op$C

LHS <- t(Pr)%*% C + time_step * t(Pl)
# Initialize U matrix to store solution at each time step
U_approx1 <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_approx1[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  # Compute the right-hand side for the second equation
  RHS <- t(Pr)%*% ( C %*% U_approx1[, k] + time_step * FF_approx[, k + 1])
  U_approx1[, k + 1] <- as.matrix(solve(LHS, RHS))
}

my_op_frac <- my.fractional.operators.frac(L, beta, C, scale.factor = kappa^2, m = m, time_step)

U_approx2 <- matrix(NA, nrow = nrow(C), ncol = length(time_seq))
U_approx2[, 1] <- U_0

# Time-stepping loop
for (k in 1:(length(time_seq) - 1)) {
  U_approx2[, k + 1] <- as.matrix(my.solver.frac(my_op_frac, my_op_frac$C %*% U_approx2[, k] + time_step * FF_approx[, k + 1]))
}

Plot

#U_approx2 <- U_approx1
# U_approx1 <- U_approx2
error.at.each.time.plotter(graph, U_true, U_approx1, U_approx2, time_seq, time_step)

graph.plotter.3d(graph, U_true, U_approx1, U_approx2, time_seq, time_step)

FF <- EIGENFUN %*% (coeff_FF * exp(-outer(EIGENVAL_ALPHA, time_seq)))
FF_true <- t(time_seq * t(FF))
graph.plotter.3d(graph, FF, FF_true, FF_approx, time_seq, time_step)